## ``` pgr_TSPeuclidean ```

• ``` pgr_TSPeuclidean ``` - Aproximation using metric algorithm.

Availability:

• Version 3.2.1

• Metric Algorithm from Boost library

• Simulated Annealing Algorithm no longer supported

• The Simulated Annealing Algorithm related parameters are ignored: max_processing_time , tries_per_temperature , max_changes_per_temperature , max_consecutive_non_changes , initial_temperature , final_temperature , cooling_factor , randomize

• Version 3.0.0

• Name change from pgr_eucledianTSP

• Version 2.3.0

• New Official function

### Description

#### Problem Definition

The travelling salesperson problem (TSP) asks the following question:

Given a list of cities and the distances between each pair of cities, which is the shortest possible route that visits each city exactly once and returns to the origin city?

#### Characteristics

• This problem is an NP-hard optimization problem.

• Metric Algorithm is used

• Implementation generates solutions that are twice as long as the optimal tour in the worst case when:

• Graph is undirected

• Graph is fully connected

• Graph where traveling costs on edges obey the triangle inequality.

• On an undirected graph:

• The traveling costs are symmetric:

• Traveling costs from ``` u ``` to ``` v ``` are just as much as traveling from ``` v ``` to ``` u ```

• Any duplicated identifier will be ignored. The coordinates that will be kept

is arbitrarly.

• The coordinates are quite similar for the same identifier, for example

```1, 3.5, 1
1, 3.499999999999 0.9999999
```
• The coordinates are quite different for the same identifier, for example

```2, 3.5, 1.0
2, 3.6, 1.1
```

### Signatures

Summary

pgr_TSPeuclidean( Coordinates SQL , ``` [start_id, end_id] ``` )
RETURNS SET OF ``` (seq, node, cost, agg_cost) ```
OR EMTPY SET
Example :

With default values

```SELECT * FROM pgr_TSPeuclidean(
\$\$
SELECT id, st_X(geom) AS x, st_Y(geom)AS y  FROM vertices
\$\$);
seq  node       cost         agg_cost
-----+------+----------------+---------------
1     1               0              0
2     6    2.2360679775   2.2360679775
3     5               1   3.2360679775
4    10   1.41421356237  4.65028153987
5     7   1.41421356237  6.06449510225
6     2   2.12132034356  8.18581544581
7     9   1.58113883008  9.76695427589
8     4             0.5  10.2669542759
9    14   1.58113883009   11.848093106
10    17   1.11803398875  12.9661270947
11    16               1  13.9661270947
12    15               1  14.9661270947
13    11   1.41421356237  16.3803406571
14    13  0.583095189485  16.9634358466
15    12  0.860232526704  17.8236683733
16     8               1  18.8236683733
17     3   1.41421356237  20.2378819357
18     1               1  21.2378819357
(18 rows)

```

### Parameters

Parameter

Type

Description

``` TEXT ```

Coordinates SQL as described below

#### TSP optional parameters

Column

Type

Default

Description

``` start_id ```

ANY-INTEGER

``` 0 ```

The first visiting vertex

• When 0 any vertex can become the first visiting vertex.

``` end_id ```

ANY-INTEGER

``` 0 ```

Last visiting vertex before returning to ``` start_vid ``` .

• When ``` 0 ``` any vertex can become the last visiting vertex before returning to ``` start_id ``` .

• When ``` NOT 0 ``` and ``` start_id = 0 ``` then it is the first and last vertex

### Inner Queries

#### Coordinates SQL

Column

Type

Description

``` id ```

``` ANY-INTEGER ```

Identifier of the starting vertex.

``` x ```

``` ANY-NUMERICAL ```

X value of the coordinate.

``` y ```

``` ANY-NUMERICAL ```

Y value of the coordinate.

### Result Columns

Returns SET OF ``` (seq, node, cost, agg_cost) ```

Column

Type

Description

seq

``` INTEGER ```

Row sequence.

node

``` BIGINT ```

Identifier of the node/coordinate/point.

cost

``` FLOAT ```

Cost to traverse from the current ``` node ``` to the next ``` node ``` in the path sequence.

• ``` 0 ``` for the last row in the tour sequence.

agg_cost

``` FLOAT ```

Aggregate cost from the ``` node ``` at ``` seq = 1 ``` to the current node.

• ``` 0 ``` for the first row in the tour sequence.

#### Test 29 cities of Western Sahara

This example shows how to make performance tests using University of Waterloo’s example data using the 29 cities of Western Sahara dataset

##### Creating a table for the data and storing the data
```CREATE TABLE wi29 (id BIGINT, x FLOAT, y FLOAT, geom geometry);
INSERT INTO wi29 (id, x, y) VALUES
(1,20833.3333,17100.0000),
(2,20900.0000,17066.6667),
(3,21300.0000,13016.6667),
(4,21600.0000,14150.0000),
(5,21600.0000,14966.6667),
(6,21600.0000,16500.0000),
(7,22183.3333,13133.3333),
(8,22583.3333,14300.0000),
(9,22683.3333,12716.6667),
(10,23616.6667,15866.6667),
(11,23700.0000,15933.3333),
(12,23883.3333,14533.3333),
(13,24166.6667,13250.0000),
(14,25149.1667,12365.8333),
(15,26133.3333,14500.0000),
(16,26150.0000,10550.0000),
(17,26283.3333,12766.6667),
(18,26433.3333,13433.3333),
(19,26550.0000,13850.0000),
(20,26733.3333,11683.3333),
(21,27026.1111,13051.9444),
(22,27096.1111,13415.8333),
(23,27153.6111,13203.3333),
(24,27166.6667,9833.3333),
(25,27233.3333,10450.0000),
(26,27233.3333,11783.3333),
(27,27266.6667,10383.3333),
(28,27433.3333,12400.0000),
(29,27462.5000,12992.2222);
```
##### Adding a geometry (for visual purposes)
```UPDATE wi29 SET geom = ST_makePoint(x,y);
```
##### Total tour cost

Getting a total cost of the tour, compare the value with the length of an optimal tour is 27603, given on the dataset

```SELECT *
FROM pgr_TSPeuclidean(\$\$SELECT * FROM wi29\$\$)
WHERE seq = 30;
seq  node      cost         agg_cost
-----+------+---------------+---------------
30     1  2266.91173136  28777.4854127
(1 row)

```
##### Getting a geometry of the tour
```WITH
tsp_results AS (SELECT seq, geom FROM pgr_TSPeuclidean(\$\$SELECT * FROM wi29\$\$) JOIN wi29 ON (node = id))
SELECT ST_MakeLine(ARRAY(SELECT geom FROM tsp_results ORDER BY seq));
st_makeline
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Visualy, The first image is the optimal solution and the second image is the solution obtained with ``` pgr_TSPeuclidean ``` .